Kerry Back
BUSI 520, Fall 2022
JGSB, Rice University
If invested funds earn a stable rate of return and funds are not withdrawn, then the account grows exponentially.
Due to exponential growth, the investment period and rate of return matter a lot!
With 1 year at 8%, \(1 \rightarrow 1.08\). After a 2nd year, we have
\[ 1.08 + (0.08 \times 1.08) = (1 \times 1.08) + (0.08 \times 1.08) \]
which is \(1.08^2\).
After \(2\) years at 8%, \(1.08^2 + 0.08\times 1.08^2 = 1.08^3\)
After \(n\) years at 8%, \(1 \rightarrow 1.08^n\).
PV factors (also called discount factors) are smaller when the horizon is longer or the rate of return is larger.
\[(1+r)^n x_0 + \cdots + (1+r)^{n-m}x_m\]
\[\frac{y_1}{1+r} + \cdots + \frac{y_n}{(1+r)^n}\]
import numpy as np
m = 10
n = 15
r = 0.08
fvFactors = (1+r)**np.arange(n, n-m-1, -1)
pvFactors = (1+r)**np.arange(-1, -n-1, -1)fvFactors are
\[(1+r)^n, \ldots (1+r)^{n-m}\]
pvFactors are
\[\frac{1}{1+r}, \ldots, \frac{1}{(1+r)^n}\]
import numpy as np
m = 4
r = 0.08
y1, y2, y3, y4 = 100, 120, 130, 140, 150
y = np.array([y1, y2, y3, y4])
pvFactors = (1+r)**np.arange(-1, -m-1, -1)
total = np.sum(y*pvFactors)